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Friday, December 5, 2014

Damped Oscillations and Forced Oscillations

Damped simple harmonic motion

Theoretically we assume that the simple
harmonic motion will always have the same time period with a constant amplitude. But here we have neglected the opposition forces like
the viscous force that are acting on the bodies that are in the state of
motion.

In real life there will be always viscous forces who are acting the motion and in general their values are directly proportional to the velocity of the bodies. Taking that into consideration a body that starts simple
harmonic motion cannot continue its simple harmonic motion forever with a
constant amplitude but it will be keep on decreasing and finally they will die
out. This kind of oscillations are called damped oscillations. We can derive a
equation for the amplitude of a damped oscillation as shown below.

It is very much clearly visible
that against the motion there is viscous force as well as the restoring force
acting and we can derive the equations basing on the Newton’s second law of
motion. It can be clearly visible that the amplitude is varying with respect to
time and the angular frequency is also decreasing because of the viscous
forces. It can be noticed that if that viscous forces ignored the angular
frequency is similar to that of the natural frequency that we had derived for
a regular simple harmonic motion.

Graph is drawn taking the time
onto the x-axis and displacement onto the y-axis and it is noticeable that as
the time progresses displacement decreases and finally the oscillation will die
down as shown below. We can also read the equation for the energy of the body
in damped simple harmonic motion which is also decreasing with respect to the
time.

Forced Oscillations and Resonance

If the body is allowed to
oscillate,it oscillates with a definite frequency depending on its
characteristic nature.This kind of oscillation is called as free oscillation
and it is going to continue to happen for ever infinite time theoretically with the
constant amplitude.This particular frequency is called as natural frequency.

But practically a body cannot
continue to vibrate with its natural
frequency forever because of the opposition so it is damped simple harmonic
motion is in real life. Anyway if you want to continue the body to be in the
simple harmonic motion with can apply an external force.This kind of a
oscillation is called as a forced oscillation.

A simple pendulum generally
vibrates with its natural frequency. If that is placed on a wooden table, then pendulum starts vibrating with its natural frequency and the wooden table also
starts vibrating with the frequency of the tuning fork. Therefore we can say
the table is under the forced vibration of the tuning fork.

We can write a mathematical
equation for a forced translation as shown below.

We can further write conditions for forced oscillations as shown below.We can not have a situation where viscous forces can be ignored practically .

In the case of viscous forces are there and if natural frequency is equal to forced frequency then the body is said to be in resonance.In that case the body will vibrate with maximum amplitude. This condition is called resonance.